To compare measures defined on different spaces, one approach is to consider metric measure spaces (abbreviated mm-spaces) which take both the metric and the measure into account together.
More precisely, it is a triple \(=(, d_,)\)  where \(\)  is a space endowed with a metric \(d_\) , and \(\in ()\) .
It is a richer model where each measure is associated to a metric on its space.
A large variety of spaces fall under this framework: shapes or graphs with their geodesic distance [1]}, Riemannian manifolds [2]} (in particular Euclidean spaces), Finsler spaces [3]}, finite dimensional Alexandrov spaces [4]}, groups equipped with the Haar measure [5]} and fractals [6]} to name a few examples.
From the ML perspective, we provide in Figure REF  illustrations of mm-spaces that are used in practice.
A first example is Euclidean point clouds in \(=^d\) , endowed with the Euclidean distance \(d_= {\cdot -\cdot }_2\)  and the uniform distribution over the samples \(=\tfrac{1}{N}\sum _i _{x_i}\) .
A second one is graphs or shapes where \(\)  is the set of nodes or a discretization of the shape, \(d_\)  is the shortest path/geodesic distance, and \(\)  is the uniform probability over the graph/shape.
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